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Blow-up and global solutions for nonlinear reaction-diffusion equations with Neumann boundary conditions. (English) Zbl 1156.35407
Summary: The type of problem under consideration is $$\cases ((1+u) \ln^\alpha(1+u))_t= \nabla\cdot(\ln^\sigma(1+u)\nabla u)+(1+u) \ln^\beta(1+u),\quad &\text{in }D \times(0,T),\\ \frac{\partial u} {\partial n}=0,\quad &\text{on }\partial D \times(0,T),\\ u(x,0)=u_0(x) >0,\quad &\text{in }\overline D,\endcases$$ where $D\subset \bbfR^N$ is a bounded domain with smooth boundary $\partial D$, $N\ge 2$. It is proved that if $\beta-1>\sigma\ge\alpha\ge 0$, the positive solution $u(x,t)$ blows up globally in $\overline D$, whereas if $0\le\beta\le \sigma\le \alpha-1$, the positive solution $u(x,t)$ is global solution. Furthermore, an upper bound of the “blow-up time”, an upper estimate of the “blow-up rate”, and an upper estimate of the global solutions are given.

35K60Nonlinear initial value problems for linear parabolic equations
35K57Reaction-diffusion equations
35K55Nonlinear parabolic equations
Full Text: DOI
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